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A285574
Irregular triangle read by rows which arises from a diagram which is similar to the diagram of A261699, but here the even-indexed zig-zag lines are located on the right-hand side of the vertical axis of the diagram.
3
1, 1, 1, 3, 1, 0, 1, 5, 1, 3, 0, 1, 0, 7, 1, 0, 0, 1, 3, 9, 1, 0, 5, 0, 1, 0, 0, 11, 1, 3, 0, 0, 1, 0, 0, 13, 1, 0, 7, 0, 1, 3, 5, 0, 15, 1, 0, 0, 0, 0, 1, 0, 0, 0, 17, 1, 3, 0, 9, 0, 1, 0, 0, 0, 19, 1, 0, 5, 0, 0, 1, 3, 0, 7, 0, 21, 1, 0, 0, 0, 11, 0, 1, 0, 0, 0, 0, 23, 1, 3, 0, 0, 0, 0, 1, 0, 5, 0, 0, 25
OFFSET
1,4
COMMENTS
In the diagram we have that:
The number of horizontal line segments in the n-th row of the structure equals A001227(n), the number of partitions of n into consecutive parts.
The number of horizontal line segments in the left-hand part of the n-th row equals A082647, the number of partitions of n into an odd number of consecutive parts.
The number of horizontal line segments in the right-hand part of the n-th row equals A131576, the number of partitions of n into an even number of consecutive parts.
The diagram explains the unusual ordering of the terms in the triangle A261699, in which the even-indexed zig-zag lines are located on the left-hand side of the vertical axis of the diagram.
EXAMPLE
Triangle begins:
1;
1;
1, 3;
1, 0,
1, 5;
1, 3, 0;
1, 0, 7;
1, 0, 0;
1, 3, 9;
1, 0, 5, 0;
1, 0, 0, 11;
1, 3, 0, 0;
1, 0, 0, 13;
1, 0, 7, 0;
1, 3, 5, 0, 15;
1, 0, 0, 0, 0;
1, 0, 0, 0, 17;
1, 3, 0, 9, 0;
1, 0, 0, 0, 19;
1, 0, 5, 0, 0;
1, 3, 0, 7, 0, 21;
...
Illustration of initial terms of the diagram:
Row _
1 _|1|
2 _|1 |_
3 _|1 |3|
4 _|1 |0|_
5 _|1 _| 5|
6 _|1 |3| 0|_
7 _|1 |0| 7|
8 _|1 _|0| 0|_
9 _|1 |3 |_ 9|
10 _|1 |0 |5| 0|_
11 _|1 _|0 |0| 11|
12 _|1 |3 |0| 0|_
13 _|1 |0 |0|_ 13|
14 _|1 _|0 _| 7| 0|_
15 _|1 |3 |5| 0| 15|
16 _|1 |0 |0| 0| 0|_
17 _|1 _|0 |0| 0|_ 17|
18 _|1 |3 |0| 9| 0|_
19 _|1 |0 _|0| 0| 19|
20 _|1 _|0 |5 |_ 0| 0|_
21 |1 |3 |0 |7| 0| 21|
...
(Compare with the diagram of A261699.)
CROSSREFS
Positive terms give A182469.
Row n has length A003056(n).
The sum of row n is A000593(n).
Column k starts in row A000217(k).
The number of positive terms in row n is A001227(n).
Sequence in context: A050143 A103495 A261699 * A354821 A081719 A327618
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Apr 21 2017
STATUS
approved